A123956 Triangle of coefficients of polynomials P(k) = 2*X*P(k - 1) - P(k - 2), P(0) = -1, P(1) = 1 + X, with twisted signs.

-1, 1, 1, -1, -2, -2, 1, -3, 4, 4, -1, 4, 8, -8, -8, 1, 5, -12, -20, 16, 16, -1, -6, -18, 32, 48, -32, -32, 1, -7, 24, 56, -80, -112, 64, 64, -1, 8, 32, -80, -160, 192, 256, -128, -128, 1, 9, -40, -120, 240, 432, -448, -576, 256, 256, -1, -10, -50, 160, 400, -672, -1120, 1024, 1280, -512, -512

Offset

0,5

Comments

Up to signs also the coefficients of polynomials y(n+1) = y(n-1) - 2*h*y(n), arising when the ODE y' = -y is numerically solved with the leapfrog (a.k.a. two-step Nyström) method, with y(0) = 1, y(1) = 1 - h. In this case, the coefficients are negative exactly for the odd powers of h. - M. F. Hasler, Nov 30 2022

References

CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

Links

Table of n, a(n) for n=0..65.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Lutterbach, Approximating ODE y′=f(t,y) by using leapfrog method, Mathematics Stack Exchange, Nov 21 2019

Formula

From M. F. Hasler, Nov 30 2022: (Start)

a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,

a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),

a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),

a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },

a(n,4) / a(n,3) = n/4 if n is even. (End)

Example

Triangle begins:
  {-1},
  { 1,   1},
  {-1,  -2,  -2},
  { 1,  -3,   4,    4},
  {-1,   4,   8,   -8,   -8},
  { 1,   5, -12,  -20,   16,   16},
  {-1,  -6, -18,   32,   48,  -32,   -32},
  { 1,  -7,  24,   56,  -80, -112,    64,   64},
  {-1,   8,  32,  -80, -160,  192,   256, -128, -128},
  { 1,   9, -40, -120,  240,  432,  -448, -576,  256,  256},
  {-1, -10, -50,  160,  400, -672, -1120, 1024, 1280, -512, -512},
  ...

Mathematica

p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
An[d_] := Table[If[n == d && m <n, -w[[n]][[d - m + 1]], If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}];
b = Table[CoefficientList[ExpandAll[y^(d - 1)*(CharacteristicPolynomial[An[d], x] /. x -> 1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten

Prog

(PARI) P=List([-1, 1-'x]); {A123956(n, k)=for(i=#P, n+1, listput(P, P[i-1]-2*'x*P[i])); polcoef(P[n+1], k)*(-1)^((n-k-1)\2+!k*n\2)} \\ M. F. Hasler, Nov 30 2022

Crossrefs

Cf. A123235.

Sequence in context: A303273 A193820 A216368 * A368606 A331953 A113594

Adjacent sequences: A123953 A123954 A123955 * A123957 A123958 A123959

Keyword

uned,tabl,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 27 2006

Extensions

Offset changed to 0 by M. F. Hasler, Nov 30 2022

Status

approved